1.Introduction M.M.El-Dessoky QualitativeBehaviorofRationalDifferenceEquationofBigOrder ResearchArticle

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Volume 2013, Article ID 495838,6pages http://dx.doi.org/10.1155/2013/495838

Research Article

Qualitative Behavior of Rational Difference Equation of Big Order

M. M. El-Dessoky

^1,2

1Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Correspondence should be addressed to M. M. El-Dessoky; [email protected] Received 4 February 2013; Accepted 20 April 2013

Academic Editor: Cengiz C¸ inar

Copyright © 2013 M. M. El-Dessoky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the global convergence, boundedness, and periodicity of solutions of the recursive sequence𝑥_𝑛+1 = (𝑎𝑥_𝑛−𝑙 + 𝑏𝑥_𝑛−𝑥)/(𝑐 + 𝑑𝑥_𝑛−𝑙𝑥_𝑛−𝑘),𝑛 = 0, 1, . . . ,where the parametersa, b, c,anddare positive real numbers, and the initial conditions 𝑥_−𝑡, 𝑥_−𝑡+1, . . . , 𝑥₋₁and𝑥₀are positive real numbers where𝑡 =max{𝑘, 𝑙}.

1. Introduction

Recently, there has been a lot of interest in studying the global attractivity, the boundedness character, and the periodicity nature of nonlinear difference equations see for example, [1–

22].

The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order; recently, many researchers have investigated the behavior of the solution of difference equations. For example, in [8]. Elabbasy et al.

investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequence:

𝑥_𝑛+1= 𝑎𝑥_𝑛− 𝑏𝑥_𝑛

𝑐𝑥_𝑛− 𝑑𝑥_𝑛−1. (1) Elabbasy et al. [9] investigated the global stability, boundedness, and periodicity character and gave the solution of some special cases of the difference equation

𝑥_𝑛+1= 𝛼𝑥_𝑛−𝑘

𝛽 + 𝛾∏^𝑘_𝑖=0𝑥_𝑛−𝑖. (2)

Elabbasy et al. [10] investigated the global stability and periodicity character and gave the solution of some special cases of the difference equation

𝑥_𝑛+1= 𝑑𝑥_𝑛−𝑙𝑥_𝑛−𝑘

𝑐𝑥_𝑛−𝑠− 𝑏 + 𝑎. (3)

Saleh and Aloqeili [23] investigated the difference equation 𝑦_𝑛+1= 𝐴 + 𝑦_𝑛

𝑦_𝑛−𝑘, with𝐴 < 0. (4) Wang et al. [24] studied the global attractivity of the equilibrium point and the asymptotic behavior of the solutions of the difference equation

𝑥_𝑛+1= 𝑎𝑥_𝑛−𝑙𝑥_𝑛−𝑘

𝛼 + 𝑏𝑥_𝑛−𝑠+ 𝑐𝑥_𝑛−𝑡. (5) In [25], Wang et al. investigated the asymptotic behavior of equilibrium point for a family of rational difference equation

𝑥_𝑛+1= ∑^𝑡_𝑖=1𝐴_𝑠_𝑖𝑥_𝑛−𝑠_𝑖

𝐵 + 𝐶∏^𝑘_𝑗=1𝑥_𝑛−𝑡_𝑗 + 𝐷𝑥_𝑛. (6) Yalc¸inkaya [26] considered the dynamics of the difference equation

𝑥_𝑛+1= 𝛼 +𝑥_𝑛−𝑚

𝑥^𝑘_𝑛 . (7)

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Zayed and El-Moneam [27,28] studied the behavior of the following rational recursive sequences:

𝑥_𝑛+1= 𝑎𝑥_𝑛− 𝑏𝑥_𝑛

𝑐𝑥_𝑛− 𝑑𝑥_𝑛−𝑘, 𝑥_𝑛+1= 𝛼 + 𝛽𝑥_𝑛+ 𝛾𝑥_𝑛−1 𝐴 + 𝐵𝑥_𝑛+ 𝐶𝑥_𝑛−1.

(8) For some related works see [29–39].

Our goal in this paper is to investigate the global stability character and the periodicity of solutions of the recursive sequence

𝑥_𝑛+1= 𝑎𝑥_𝑛−𝑙+ 𝑏𝑥_𝑛−𝑘

𝑐 + 𝑑𝑥_𝑛−𝑙𝑥_𝑛−𝑘, 𝑛 = 0, 1, . . . , (9) where the parameters𝑎, 𝑏, 𝑐,and𝑑are positive real numbers and the initial conditions𝑥_−𝑡, 𝑥_−𝑡+1, . . . , 𝑥₋₁ and𝑥₀ are positive real numbers where𝑡 =max{𝑘, 𝑙}.

2. Local Stability of the Equilibrium Point of (9)

This section deals with the local stability character of the equilibrium point of (9)

Equation (9) has equilibrium points given by 𝑥 = (𝑎 + 𝑏) 𝑥

𝑐 + 𝑑𝑥² , (10)

then

𝑥 {𝑑𝑥²+ 𝑐 − 𝑎 − 𝑏} = 0. (11) Then the equilibrium points of (9) are given by

𝑥 = 0 or 𝑥 = √𝑎 + 𝑏 − 𝑐

𝑑 when𝑎 + 𝑏 > 𝑐. (12) Let𝑓 : (0, ∞)² → (0, ∞)be a continuously differentiable function defined by

𝑓 (𝑢,V) = 𝑎𝑢 + 𝑏V

𝑐 + 𝑑𝑢V. (13)

Therefore, it follows that

𝜕𝑓 (𝑢,V)

𝜕𝑢 = 𝑎𝑐 − 𝑏𝑑V²

(𝑐 + 𝑑𝑢V)², 𝜕𝑓 (𝑢,V)

𝜕V = 𝑏𝑐 − 𝑎𝑑𝑢² (𝑐 + 𝑑𝑢V)². (14) Theorem 1. The following statements are true.

(1)If𝑎 + 𝑏 ≤ 𝑐, then the only equilibrium point𝑥 = 0of (9)is locally stable.

(2)If𝑎 + 𝑏 > 𝑐, then the positive equilibrium point𝑥 =

√(𝑎 + 𝑏 − 𝑐)/𝑑of(9)is locally stable if|𝑐−𝑏|+|𝑐−𝑎| <

𝑎 + 𝑏.

Proof. (1)If𝑎 + 𝑏 ≤ 𝑐, then we see from (14) that

𝜕𝑓 (0, 0)

𝜕𝑥_𝑛−𝑙 = 𝑎

𝑐, 𝜕𝑓 (0, 0)

𝜕𝑥_𝑛−𝑘 =𝑏

𝑐. (15)

Then, the linearized equation associated with (9) about𝑥 = 0 is

𝑦_𝑛+1−𝑎 𝑐𝑦_𝑛−𝑙−𝑏

𝑐𝑦_𝑛−𝑘= 0, (16) whose characteristic equation is

𝜆^𝑘+1−𝑎 𝑐𝜆^𝑘−𝑙−𝑏

𝑐 = 0. (17)

Then, (16) is asymptotically stable if𝑎 + 𝑏 < 𝑐, and then the equilibrium point𝑥 = 0of (9) is locally stable.

(2)If 𝑎 + 𝑏 > 𝑐, then we see from (14) that

𝜕𝑓 (𝑥, 𝑥)

𝜕𝑥_𝑛−𝑙 = 𝑎𝑐 − 𝑏𝑑 ((𝑎 + 𝑏 − 𝑐) /𝑑) (𝑐 + 𝑑 ((𝑎 + 𝑏 − 𝑐)/𝑑))² = 𝑐 − 𝑏

𝑎 + 𝑏,

𝜕𝑓 (𝑥, 𝑥)

𝜕𝑥_𝑛−𝑘 = 𝑏𝑐 − 𝑎𝑑 ((𝑎 + 𝑏 − 𝑐) /𝑑) (𝑐 + 𝑑 ((𝑎 + 𝑏 − 𝑐)/𝑑))² = 𝑐 − 𝑎

𝑎 + 𝑏. (18)

Then, the linearized equation of (9) about𝑥is 𝑦_𝑛+1− 𝑐 − 𝑏

𝑎 + 𝑏𝑦_𝑛−𝑙− 𝑐 − 𝑎

𝑎 + 𝑏𝑦_𝑛−𝑘= 0, (19) whose characteristic equation is

𝜆^𝑘+1−𝑐 − 𝑏

𝑎 + 𝑏𝜆^𝑘−𝑙− 𝑐 − 𝑎

𝑎 + 𝑏 = 0. (20)

Then, (19) is asymptotically stable if all roots of (20) lie in the open disc|𝜆| < 1, that is, if

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑐 − 𝑏

𝑎 + 𝑏󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑐 − 𝑎

𝑎 + 𝑏󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < 1, (21) which is true if

|𝑐 − 𝑏| + |𝑐 − 𝑎| < 𝑎 + 𝑏. (22) The proof is complete.

3. Boundedness of the Solutions of (9)

Here, we study the boundedness nature of the solutions of (9).

Theorem 2. Every solution of (9)is bounded if𝑐 > 𝑎 + 𝑏.

Proof. Let{𝑥_𝑛}^∞_𝑛=−𝑡be a solution of (9). It follows from (9) that 𝑥_𝑛+1= 𝑎𝑥_𝑛−𝑙+ 𝑏𝑥_𝑛−𝑘

𝑐 + 𝑑𝑥_𝑛−𝑙𝑥_𝑛−𝑘 ≤𝑎𝑥_𝑛−𝑙+ 𝑏𝑥_𝑛−𝑘

𝑐 . (23)

By using a comparison, we can write the right-hand side as follows:

𝑦_𝑛+1= 𝑎𝑦_𝑛−𝑙 𝑐 +𝑏𝑦_𝑛−𝑘

𝑐 , (24)

and this equation is locally asymptotically stable if𝑎 + 𝑏 < 𝑐 and converges to the equilibrium point𝑦 = 0.

Therefore,

lim sup

𝑛 → ∞ 𝑥_𝑛 = 0. (25)

Thus, the solution is bounded.

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4. Existence of Periodic Solutions

In this section, we study the existence of periodic solutions of (9). The following theorem states the necessary and sufficient conditions that this equation has periodic solutions of prime period two.

Theorem 3. Equation(9)has a prime period two solutions if and only if one of the following statements holds:

(1)𝑎 + 𝑏 − 𝑐 > 0, and l, k—odd, (2)𝑎 + 𝑐 − 𝑏 > 0, and k—odd, l—even, (3)𝑏 + 𝑐 − 𝑎 > 0, and l—odd, k—even.

Proof. We will prove the theorem when condition(1)is true, and the proof of the other cases is similar and so we will be omit it.

First suppose that there exists a prime period two solution . . . , 𝑝, 𝑞, 𝑝, 𝑞, . . . , (26) of (9). We will prove that Condition(1)holds.

We see from (9) that 𝑝 = (𝑎 + 𝑏) 𝑝

𝑐 + 𝑑𝑝² , 𝑞 = (𝑎 + 𝑏) 𝑞

𝑐 + 𝑑𝑞² . (27) Then,

𝑐 + 𝑑𝑝²= 𝑎 + 𝑏, (28)

𝑐 + 𝑑𝑞²= 𝑎 + 𝑏. (29)

Subtracting (28) from (29) gives

𝑑 (𝑝²− 𝑞²) = 0. (30)

Since𝑝 ̸= 𝑞, it follows that

𝑝 = −𝑞. (31)

Again, from (28) and (29)

𝑝²= 𝑞²= 𝑎 + 𝑏 − 𝑐

𝑑 , (32)

and so

𝑎 + 𝑏 − 𝑐 > 0. (33)

Therefore, inequality(1)holds.

Second, suppose that inequality(1)is true. We will show that (9) has a prime period two solution.

Assume that

𝑝 = +√𝑎 + 𝑏 − 𝑐

𝑑 , 𝑞 = −√𝑎 + 𝑏 − 𝑐

𝑑 . (34)

We see from inequality(1)that

𝑎 + 𝑏 − 𝑐 > 0. (35)

Therefore,𝑝and𝑞are distinct real numbers.

Set

𝑥_−𝑙= 𝑥_−𝑘= 𝑝, 𝑥₋₂= 𝑞, 𝑥₋₁= 𝑝, 𝑥₀= 𝑞.

(36) We wish to show that

𝑥₁= 𝑥₋₁ = 𝑝, 𝑥₂= 𝑥₀= 𝑞. (37) It follows from (9) that

𝑥₁=(𝑎 + 𝑏) 𝑝

𝑐 + 𝑑𝑝² = (𝑎 + 𝑏) √(𝑎 + 𝑏 − 𝑐) /𝑑

𝑐 + 𝑑 ((𝑎 + 𝑏 − 𝑐) /𝑑) = √𝑎 + 𝑏 − 𝑐

𝑑 = 𝑝.

(38) Similarly, we see that

𝑥₂= (𝑎 + 𝑏) 𝑞

𝑐 + 𝑑𝑞² = − (𝑎 + 𝑏) √(𝑎 + 𝑏 − 𝑐) /𝑑 𝑐 + 𝑑 ((𝑎 + 𝑏 − 𝑐) /𝑑)

= −√𝑎 + 𝑏 − 𝑐 𝑑 = 𝑞.

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Then, it follows by induction that

𝑥_2𝑛= 𝑞, 𝑥_2𝑛+1= 𝑝, ∀𝑛 ≥ −1. (40) Thus, (9) has the prime period two solution

. . . , 𝑝, 𝑞, 𝑝, 𝑞, . . . , (41) where𝑝and𝑞are distinct roots of a quadratic equation, and the proof is complete.

5. Global Attractor of the Equilibrium Point of (9)

In this section, we investigate the global asymptotic stability of (9). If we take the function𝑓(𝑢,V)defined by (16), then we have four cases of the monotonicity behavior in its arguments (all of these cases we suppose that𝑎 + 𝑏 > 𝑐).

Theorem 4. If the function𝑓(𝑢,V)defined by(16)is nonde- creasing (or nonincreasing) in𝑢,V, then the positive equilib- rium point𝑥 = √(𝑎 + 𝑏 − 𝑐)/𝑑is a global attractor of(9).

Proof. Let{𝑥_𝑛}^∞_𝑛=−𝑡be a solution of (9) and again let𝑓be a function defined by (16).

We will prove the theorem when𝑓(𝑢,V)is nondecreasing and the proof of the other cases is similar, and so we will omit it.

Suppose that(𝑚, 𝑀)is a solution of the systems 𝑀 = 𝑓(𝑀, 𝑀)and 𝑚 = 𝑔(𝑚, 𝑚). Then, from (9), we see that

𝑀 = 𝑎𝑀 + 𝑏𝑀

𝑐 + 𝑑𝑀² , 𝑚 =𝑎𝑚 + 𝑏𝑚

𝑐 + 𝑑𝑚² , (42) or

𝑐 + 𝑑𝑀²= 𝑎 + 𝑏, 𝑐 + 𝑑𝑚²= 𝑎 + 𝑏. (43)

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Subtracting these two equations, we obtain

𝑑 (𝑀 − 𝑚) (𝑀 + 𝑚) = 0. (44) Under the condition𝑑 > 0, we see that

𝑀 = 𝑚. (45)

It follows byTheorem 2that𝑥is a global attractor of (9), and then the proof is complete.

Theorem 5. If the function 𝑓(𝑢,V)defined by (16) is non- decreasing in 𝑢 and nonincreasing in V, then the positive equilibrium point𝑥 = √(𝑎 + 𝑏 − 𝑐)/𝑑is a global attractor of (9)if𝑐 + 𝑏 > 𝑎.

Proof. Let{𝑥_𝑛}^∞_𝑛=−𝑡be a solution of (9) and again let𝑓be a function defined by (16).

Suppose that(𝑚, 𝑀)is a solution of the systems 𝑀 = 𝑓(𝑀, 𝑚)and 𝑚 = 𝑔(𝑚, 𝑀). Then, from (9), we see that

𝑀 = 𝑎𝑀 + 𝑏𝑚

𝑐 + 𝑑𝑚𝑀, 𝑚 = 𝑎𝑚 + 𝑏𝑀

𝑐 + 𝑑𝑚𝑀, (46) or

𝑐𝑀 + 𝑑𝑚𝑀²= 𝑎𝑀 + 𝑏𝑚,

𝑐𝑚 + 𝑑𝑀𝑚²= 𝑎𝑚 + 𝑏𝑀. (47)

Subtracting these two equations, we obtain

𝑐 (𝑀 − 𝑚) + 𝑑𝑀𝑚 (𝑀 − 𝑚) = (𝑎 − 𝑏) (𝑀 − 𝑚) , (𝑀 − 𝑚) {𝑐 + 𝑏 − 𝑎 + 𝑑𝑀𝑚} = 0. (48) Under the condition𝑐 + 𝑏 > 𝑎,we see that

𝑀 = 𝑚. (49)

It follows byTheorem 2that𝑥is a global attractor of (9), and then the proof is complete.

Theorem 6. If the function𝑓(𝑢,V)defined by(16)is nonde- creasing inV,nonincreasing in𝑢. Then the positive equilibrium point𝑥 = √(𝑎 + 𝑏 − 𝑐)/𝑑is a global attractor of(9)if𝑐+𝑎 > 𝑏.

Proof. The proof is similar to the previous Theorem and so we will be omit it.

Lemma 7. When𝑐 ≥ 𝑎+𝑏then the equilibrium point𝑥 = 0of (9)is global attractor.

Proof. If𝑐 ≥ 𝑎 + 𝑏, then the proof follows byTheorem 2.

6. Numerical Examples

For confirming the results of this paper, we consider numerical examples which represent different types of solutions to (9).

Example 1. We assume that𝑙 = 1, 𝑘 = 2, 𝑥₋₂ = 3, 𝑥₋₁ = 2, 𝑥₀= 6, 𝑎 = 2, 𝑏 = 5, 𝑐 = 8,and 𝑑 = 6. SeeFigure 1.

0 5 10 15 20 25 30 35 40

0 1 2 3 4 5 6

𝑥(𝑛)

𝑛

Figure 1: It shows the solution of (9) with𝑙 = 1, 𝑘 = 2, 𝑥₋₂ = 3, 𝑥₋₁= 2, 𝑥₀= 6, 𝑎 = 2, 𝑏 = 5, 𝑐 = 8,and𝑑 = 6.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6

𝑥(𝑛)

𝑛

Figure 2: It shows the behavior of the solution of (9) with𝑙 = 1, 𝑘 = 3, 𝑥₋₃ = 3, 𝑥₋₂ = 1, 𝑥₋₁ = 6, 𝑥₀ = 5, 𝑎 = 9, 𝑏 = 13, 𝑐 = 0.1,and𝑑 = 2.

0 2 4 6 8 10 12 14 16 18 20

0 2 4

𝑛

−2

−4

𝑥(𝑛)

Figure 3: It shows the periodicity of the solution of (9) when𝑙 = 3, 𝑘 = 1, 𝑥₋₃ = 𝑥₋₁ = 𝑝, 𝑥₋₂ = 𝑥₀ = 𝑞, 𝑎 = 9, 𝑏 = 13, 𝑐 = 0.1,and𝑑 = 2.

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0 2 4 6 8 10 12 14 16 18 20 0

1 2

𝑥(𝑛)

𝑛

−2

−1

Figure 4: It shows the periodicity of the solution of (9) when𝑙 = 4, 𝑘 = 3, 𝑥₋₃= 𝑥₋₁= 𝑞, 𝑥₋₄= 𝑥₋₂ = 𝑥₀= 𝑝, 𝑎 = 9, 𝑏 = 5, 𝑐 = 3,and𝑑 = 2.

Example 2. SeeFigure 2, since𝑙 = 1, 𝑘 = 3, 𝑥₋₃ = 3, 𝑥₋₂ = 1, 𝑥₋₁ = 6, 𝑥₀= 5, 𝑎 = 9, 𝑏 = 13, 𝑐 = 0.1, 𝑑 = 2.

Example 3. Figure 3shows the solutions when𝑙 = 3, 𝑘 = 1, 𝑥₋₃ = 𝑥₋₁ = 𝑝, 𝑥₋₂ = 𝑥₀ = 𝑞, 𝑎 = 9, 𝑏 = 13, 𝑐 = 0.1,and𝑑 = 2.(Since𝑝, 𝑞 = ±√(𝑎 + 𝑏 − 𝑐)/𝑑).

Example 4. Figure 4 shows the solutions when𝑙 = 4, 𝑘 = 3, 𝑥₋₃ = 𝑥₋₁ = 𝑞, 𝑥₋₄ = 𝑥₋₂ = 𝑥₀ = 𝑝, 𝑎 = 9, 𝑏 = 5, 𝑐 = 3,and𝑑 = 2.(Since𝑝, 𝑞 = ±√(𝑎 + 𝑏 − 𝑐)/𝑑).

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no.

(130-056-D1433). The author, therefore, acknowledges with thanks to DSR technical and financial support.

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1.Introduction M.M.El-Dessoky QualitativeBehaviorofRationalDifferenceEquationofBigOrder ResearchArticle

Research Article

Qualitative Behavior of Rational Difference Equation of Big Order

M. M. El-Dessoky

1. Introduction

2. Local Stability of the Equilibrium Point of (9)

3. Boundedness of the Solutions of (9)

4. Existence of Periodic Solutions

5. Global Attractor of the Equilibrium Point of (9)

6. Numerical Examples

Acknowledgments

References

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Stochastic Analysis